Comments on finite termination of the generalized Newton method for absolute value equations

We consider the generalized Newton method (GNM) for the absolute value equation (AVE) $Ax-|x|=b$. The method has finite termination property whenever it is convergent, no matter whether the AVE has a unique solution. We prove that GNM is convergent whenever $\rho(|A^{-1}|)<1/3$. We also present new results for the case where $A-I$ is a nonsingular $M$-matrix or an irreducible singular $M$-matrix. When $A-I$ is an irreducible singular $M$-matrix, the AVE may have infinitely many solutions. In this case, we show that GNM always terminates with a uniquely identifiable solution, as long as the initial guess has at least one nonpositive component

The Unique Solvability Conditions for the Generalized Absolute Value Equations

This paper investigates the conditions that guarantee unique solvability and unsolvability for the generalized absolute value equations (GAVE) given by $Ax - B \vert x \vert = b$. Further, these conditions are also valid to determine the unique solution of the generalized absolute value matrix equations (GAVME) $AX - B \vert X \vert =F$. Finally, certain aspects related to the solvability and unsolvability of the absolute value equations (AVE) have been deliberated upon

Necessary and sufficient conditions for unique solvability of absolute value equations: A Survey

In this survey paper, we focus on the necessary and sufficient conditions for the unique solvability and unsolvability of the absolute value equations (AVEs) during the last twenty years (2004 to 2023). We discussed unique solvability conditions for various types of AVEs like standard absolute value equation (AVE), Generalized AVE (GAVE), New generalized AVE (NGAVE), Triple AVE (TAVE) and a class of NGAVE based on interval matrix, P-matrix, singular value conditions, spectral radius and $\mathcal{W}$-property. Based on the unique solution of AVEs, we also discussed unique solvability conditions for linear complementarity problems (LCP) and horizontal linear complementarity problems (HLCP)

Improved Harmony Search Algorithm with Chaos for Absolute Value Equation

 In this paper, an improved harmony search with chaos (HSCH) is presented for solving NP-hard absolute value equation (AVE) Ax - |x| = b, where A is an arbitrary square matrix whose singular values exceed one. The simulation results in solving some given AVE problems demonstrate that the HSCH algorithm is valid and outperforms the classical HS algorithm (HS) and HS algorithm with differential mutation operator (HSDE)